Verhulst model with Lévy white noise excitation
Radiophysics Department, Nizhniy Novgorod State University, 23 Gagarin Ave., 603950 Nizhniy Novgorod, Russia
2 Dipartimento di Fisica e Tecnologie Relative, Group of Interdisciplinary Physics (URL: ) , Università di Palermo and CNISM-INFM, Unità di Palermo, Viale delle Scienze, 90128 Palermo, Italy
Revised: 27 June 2008
Published online: 5 September 2008
The transient dynamics of the Verhulst model perturbed by arbitrary non-Gaussian white noise is investigated. Based on the infinitely divisible distribution of the Lévy process we study the nonlinear relaxation of the population density for three cases of white non-Gaussian noise: (i) shot noise; (ii) noise with a probability density of increments expressed in terms of Gamma function; and (iii) Cauchy stable noise. We obtain exact results for the probability distribution of the population density in all cases, and for Cauchy stable noise the exact expression of the nonlinear relaxation time is derived. Moreover starting from an initial delta function distribution, we find a transition induced by the multiplicative Lévy noise, from a trimodal probability distribution to a bimodal probability distribution in asymptotics. Finally we find a nonmonotonic behavior of the nonlinear relaxation time as a function of the Cauchy stable noise intensity.
PACS: 05.40.Fb – Random walks and Levy flights / 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion / 02.50.-r – Probability theory, stochastic processes, and statistics / 87.23.Cc – Population dynamics and ecological pattern formation
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2008